📄 8th Grade Math (Algebra I): Create and Solve Linear Equations and Inequalities in One Variable with Real-Life Applications Worksheet
📌 1. True / False
1. A linear equation always has exactly one solution.
2. When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed.
3. The expression \( 4x + 5 = 17 \) is an example of a linear inequality.
4. An inequality like \( x > 10 \) represents all numbers greater than 10.
5. Real-life problems involving a constant rate of change can often be modeled using linear equations or inequalities.
✏️ 2. Fill in the Blanks
1. An equation states that two mathematical expressions are .
2. To solve for a variable in a linear equation, we use operations to isolate the variable.
3. The solution to an inequality is often represented by an of numbers, rather than a single value.
4. When translating a real-life problem into an algebraic expression, identifying the is crucial.
5. A linear equation in one variable will have a maximum of solution(s).
🔗 3. Matching
« An algebraic statement that two expressions are equal, involving a variable raised to the power of one.
« A mathematical statement comparing two expressions using symbols like \(<, >, \le, \ge \).
« A symbol, usually a letter, representing an unknown quantity.
« The collection of all values that make an equation or inequality true.
« Operations that undo each other, used to isolate a variable in an equation or inequality.
✍️ 4. Short Answer Questions
1. Explain why checking your solution is an important step when solving equations.
💡 Suggested Answer: Checking your solution is important to ensure that the value you found makes the original equation true. This helps to catch any calculation errors and confirms the accuracy of your answer.
2. Give an example of a real-life scenario that could be modeled by the inequality \( x \le 60 \).
💡 Suggested Answer: A common example is a speed limit. If the speed limit on a highway is 60 mph, then a car's speed \( x \) must be less than or equal to 60 mph.
🎯 5. Multiple Choice
1. Solve the equation: \( 4x - 9 = 15 \)
2. Which inequality represents the statement: "A number increased by 7 is at most 20"?
3. A taxi charges a flat fee of USD 2.50 plus USD 1.50 per mile. If a ride cost a total of USD 10.00, how many miles was the ride?
📝 6. Open-Ended Questions
1. Solve the equation: \( 3(2x - 5) + 7 = 22 \)
💡 Solution Steps:
To solve the equation \( 3(2x - 5) + 7 = 22 \):
Step 1: Distribute the 3 to the terms inside the parentheses.
\( 6x - 15 + 7 = 22 \)
Step 2: Combine the constant terms on the left side.
\( 6x - 8 = 22 \)
Step 3: Add 8 to both sides of the equation to isolate the term with \( x \).
\( 6x - 8 + 8 = 22 + 8 \)
\( 6x = 30 \)
Step 4: Divide both sides by 6 to solve for \( x \).
\( \frac{6x}{6} = \frac{30}{6} \)
\( x = 5 \)
Therefore, the solution is \( x = 5 \).
2. A local gym charges a one-time enrollment fee of USD 50 and then USD 30 per month. If a member paid a total of USD 290, for how many months did they sign up?
💡 Solution Steps:
Let \( m \) represent the number of months the member signed up for.
Step 1: Set up the equation based on the given information.
The total cost is the enrollment fee plus the monthly fee multiplied by the number of months.
\( 50 + 30m = 290 \)
Step 2: Subtract the enrollment fee from both sides of the equation.
\( 50 + 30m - 50 = 290 - 50 \)
\( 30m = 240 \)
Step 3: Divide both sides by 30 to solve for \( m \).
\( \frac{30m}{30} = \frac{240}{30} \)
\( m = 8 \)
Therefore, the member signed up for 8 months.
3. Emily wants to save at least USD 300 to buy a new tablet. She already has USD 80 saved and plans to save USD 25 each week. Write and solve an inequality to find the minimum number of weeks Emily needs to save.
💡 Solution Steps:
Let \( w \) represent the number of weeks Emily needs to save.
Step 1: Write an inequality to represent the situation.
Emily's current savings plus her weekly savings must be at least USD 300.
\( 80 + 25w \ge 300 \)
Step 2: Subtract 80 from both sides of the inequality.
\( 80 + 25w - 80 \ge 300 - 80 \)
\( 25w \ge 220 \)
Step 3: Divide both sides by 25 to solve for \( w \).
\( \frac{25w}{25} \ge \frac{220}{25} \)
\( w \ge 8.8 \)
Since Emily can only save in whole weeks, she needs to save for at least 9 weeks to reach her goal. Saving for 8 weeks would only give her \( 80 + 25(8) = 80 + 200 = 280 \) USD, which is not enough. Therefore, she needs 9 weeks.
Name Surname: .................................. Date: .... / .... / 202...
Create and Solve Linear Equations and Inequalities in One Variable with Real-Life Applications Worksheet
SCORE
A. True (T) / False (F)
( .... )
A linear equation always has exactly one solution.
( .... )
When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed.
( .... )
The expression \( 4x + 5 = 17 \) is an example of a linear inequality.
( .... )
An inequality like \( x > 10 \) represents all numbers greater than 10.
( .... )
Real-life problems involving a constant rate of change can often be modeled using linear equations or inequalities.
B. Fill in the Blanks
1)
An equation states that two mathematical expressions are .....................
2)
To solve for a variable in a linear equation, we use .................... operations to isolate the variable.
3)
The solution to an inequality is often represented by an .................... of numbers, rather than a single value.
4)
When translating a real-life problem into an algebraic expression, identifying the .................... is crucial.
5)
A linear equation in one variable will have a maximum of .................... solution(s).
C. Matching Concepts
( .... )
An algebraic statement that two expressions are equal, involving a variable raised to the power of one.
- Variable
( .... )
A mathematical statement comparing two expressions using symbols like \(<, >, \le, \ge \).
- Linear Equation
( .... )
A symbol, usually a letter, representing an unknown quantity.
- Inverse Operations
( .... )
The collection of all values that make an equation or inequality true.
- Solution Set
( .... )
Operations that undo each other, used to isolate a variable in an equation or inequality.
- Inequality
D. Short Answer Questions
1)
Explain why checking your solution is an important step when solving equations.
2)
Give an example of a real-life scenario that could be modeled by the inequality \( x \le 60 \).
E. Multiple Choice Questions
1)
Solve the equation: \( 4x - 9 = 15 \)
A) 4B) 6C) 3D) 24
2)
Which inequality represents the statement: "A number increased by 7 is at most 20"?
A) \( n + 7 < 20 \)B) \( n + 7 \le 20 \)C) \( n + 7 > 20 \)D) \( n + 7 \ge 20 \)
3)
A taxi charges a flat fee of USD 2.50 plus USD 1.50 per mile. If a ride cost a total of USD 10.00, how many miles was the ride?
A) 4 milesB) 5 milesC) 6 milesD) 7 miles
F. Open-Ended Questions
1)
Solve the equation: \( 3(2x - 5) + 7 = 22 \)
2)
A local gym charges a one-time enrollment fee of USD 50 and then USD 30 per month. If a member paid a total of USD 290, for how many months did they sign up?
3)
Emily wants to save at least USD 300 to buy a new tablet. She already has USD 80 saved and plans to save USD 25 each week. Write and solve an inequality to find the minimum number of weeks Emily needs to save.